Differential forms and κ-Minkowski spacetime from extended twist

Differential forms and κ-Minkowski spacetime from extended twist Tajron Juri´c 1 Rudjer Boškovi´c Institute, Bijeniˇcka c.54, HR-10002 Zagreb, Croatia Stjepan Meljanac 2 ,

arXiv:1211.6612v4 [hep-th] 5 Jul 2013

Rudjer Boškovi´c Institute, Bijeniˇcka c.54, HR-10002 Zagreb, Croatia Rina Štrajn 3 , Jacobs University Bremen, 28759 Bremen, Germany We analyze bicovariant differential calculus on κ-Minkowski spacetime. It is shown that corresponding Lorentz generators and noncommutative coordinates compatible with bicovariant calculus cannot be realized in terms of commutative coordinates and momenta. Furthermore, κ-Minkowski space and NC forms are constructed by twist related to a bicrossproduct basis. It is pointed out that the consistency condition is not satisfied. We present the construction of κ-deformed coordinates and forms (super-Heisenberg algebra) using extended twist. It is compatible with bicovariant differential calculus with κ-deformed igl(4)-Hopf algebra. The extended twist leading to κ-Poincaré-Hopf algebra is also discussed. Keywords: noncommutative space, κ-Minkowski spacetime, differential forms, super-Heisenberg algebra, realizations, twist.

I.

INTRODUCTION

The structure of spacetime at very high energies (Planck scale lengths) is still unknown and it is believed that, at these energies, gravity effects become significant and we need to abandon the notion of smooth and continuous spacetime. Among many attempts to find a suitable model for unifying quantum field theory and gravity, one of the ideas that emerged is that of noncommutative spaces [1]-[5]. Authors inclined to this idea have followed different approaches and considered different types of noncommutative (NC) spaces, where the concept of invoking twisted Poincaré symmetry of the algebra of functions on a Minkowski spacetime using twist operator is the most elaborated one [6]. In formulating field theories on NC spaces, differential calculus plays an essential role. The requirement that this differential calculus is bicovariant and also covariant under the expected group of symmetries leads to some problems. 1 2 3

e-mail:e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]

1

One widely researched type of NC space, which is also the object of consideration in this letter, is the κ-Minkowski space [7]-[48]. This space is a Lie algebra type of deformation of the Minkowski spacetime and here the deformation parameter κ is usually interpreted as the Planck mass or the quantum gravity scale. κ-Minkowski space is also related to doubly special relativity [28]-[31]. For each NC space there is a corresponding symmetry algebra. In the case of κ-Minkowski space, the symmetry algebra is a deformation of the Poincaré algebra, known as the κ-Poincaré algebra. The κ-Poincaré algebra is also an example of a Hopf algebra. Some of the results of pursuing this line of research are, e.g., the construction of quantum field theories [32]-[39], electrodynamics [40]-[42], considerations of quantum gravity effects [43]-[45] and the modification of particle statistics [46]-[48] on κ-Minkowski space. Regarding the problem of differential calculus on κ-Minkowski space, Sitarz has shown [11] that in order to obtain bicovariant differential calculus, which is also Lorentz covariant, one has to introduce an extra cotangent direction. While Sitarz considered 3+1 dimensional space (and developed five dimensional differential calculus), Gonera et al. generalized this work to n dimensions in Ref. [12]. Another attempt to deal with this issue was made in [20] by the Abelian twist deformation of U[igl(4, R)]. Bu et al. in [21] extended the Poincaré algebra with the dilatation operator and constructed a four dimensional differential algebra on the κ-Minkowski space using a Jordanian twist of the Weyl algebra. Differential algebras of classical dimensions were also constructed in [18] and [19], from the action of a deformed exterior derivative. In [26] the authors have constructed two families of differential algebras of classical dimensions on the κ-Minkowski space, using realizations of the generators as formal power series in a Weyl superalgebra. In this approach, the realization of the Lorentz algebra is also modified, with the addition of Grassmann-type variables. As a consequence, generators of the Lorentz algebra act covariantly on one-forms, without the need to introduce an extra cotangent direction. The action is also covariant if restricted to the κ-Minkowski space. However, one loses Lorentz covariance when considering forms of order higher than one. Our motivation in this letter is to unify κ-Minkowski spacetime, κ-Poincare algebra and differential forms. We embed them into κ-deformed super-Heisenberg algebra related to bicrossproduct basis. Using extended twist, we construct a smooth mapping between κ-deformed super-Heisenberg algebra and superHeisenberg algebra. We present an extended realization for κ-deformed coordinates, Lorentz generators, and exterior derivative compatible with Lorentz covariance condition. In section II, super-Heisenberg algebra is described. In section III, realization of κ-Minkowski space and κ-Poincare algebra related to bicrossproduct basis is given. In section IV, bicovariant differential calculus is analyzed. It is pointed out that there does not exist a realization of Lorentz generators and NC coordinates compatible with bicovariant calculus in terms of commutative coordinates and momenta. In section V, κ-Minkowski space and NC forms are constructed by twist related to bicrossproduct basis. It is 2

shown that the consistency condition is not satisfied. In section VI, we present our main construction of κdeformed super-Heisenberg algebra using extended twist. Extended realizations for Lorentz generators and exterior derivative invariant under igl(4)-Hopf algebra are presented. Finally, in section VII we outline the construction of Lorentz generators, exterior derivative and one-forms for bicovariant calculus compatible with κ-Poincaré-Hopf algebra.

II.

SUPER-HEISENBERG ALGEBRA

In the undeformed case we consider spacetime coordinates xµ , derivatives ∂µ ≡ and Grassmann derivatives qµ ≡

∂ ∂ξµ

∂ ∂xµ ,

one forms dxµ ≡ ξµ ,

satisfying the following (anti)commutation relations:

[xµ , xν ] = [∂µ , ∂ν ] = 0,

[∂µ , xν ] = ηµν ,

{ξµ , ξν } = {qµ , qν } = 0,

{ξµ , qν } = ηµν ,

(1)

[xµ , ξν ] = [xµ , qν ] = [∂µ , ξν ] = [∂µ , qν ] = 0, where µ = {0, 1, 2, 3} and ηµν = diag(−1, 1, 1, 1). The algebra in (1) generates the undeformed superHeisenberg algebra SH(x, ∂, ξ, q) i.e. superphase space. The exterior derivative is defined as d = ξα ∂α , so ξµ = [d, xµ ]. We define the action ⊲ : SH(x, ∂, ξ, q) 7→ SA(x, ξ), where SA(x, ξ) ⊂ SH(x, ∂, ξ, q). The superHeisenberg algebra SH(x, ∂, ξ, q) can be written as SH = SA ST , where ST (∂, q) ⊂ SH(x, ∂, ξ, q). For any element f (x, ξ) ∈ SA(x, ξ) we have xµ ⊲ f (x, ξ) = xµ f (x, ξ), ∂µ ⊲ f (x, ξ) =

∂f , ∂xµ

ξµ ⊲ f (x, ξ) = ξµ f (x, ξ), qµ ⊲ f (x, ξ) =

∂f . ∂ξ µ

(2)

The coalgebra structure of ST (∂, q) is defined by undeformed coproducts: ∆0 ∂µ = ∂µ ⊗ 1+1 ⊗ ∂µ ,

∆0 qµ = qµ ⊗ 1 + (−)deg ⊗ qµ ,

(3)

α

deg = ξα q (mod2). The coalgebra structure with antipode and counit is (undeformed) super-Hopf algebra. Let us mention that super-Heisenberg algebra SH has also super-Hopf-algebroid structure, which will be elaborated separately. The Hopf-algebroid structure of Heisenberg algebra was discussed in [52] 4 . Now we introduce Lorentz generators Mµν : [Mµν , Mλρ ] = ηνλ Mµρ − ηµλ Mνρ − ηνρ Mµλ + ηµρ Mνλ , 4

for Hopf-algebroid structure also see [49], [50] and [27]

3

(4)

with the following undeformed coproduct: ∆0 Mµν = Mµν ⊗ 1 + 1 ⊗ Mµν

(5)

and action ⊲ : Mµν ⊲ xλ = ηνλ xµ −ηµλ xν ,

Mµν ⊲ ξλ = ηνλ ξµ − ηµλ ξν .

(6)

Mµν ⊲ 1 = 0 Using (5) and (6) we can derive the commutation relations [Mµν , xλ ] = ηνλ xµ − ηµλ xν ,

[Mµν, ξλ ] = ηνλ ξµ − ηµλ ξν ,

(7)

so that xµ and ξµ transform as vectors (the same holds for ∂µ and qµ ). The Lorentz covariance condition
Mµν ⊲ f (x, ξ)g(x, ξ) = m0 ∆0 Mµν ⊲ f ⊗ g

(8)

Mµν ⊲ d f (x, ξ) = d(Mµν ⊲ f (x, ξ))

(where m0 is the multiplication map) implies [Mµν , d] = 0,

(9)

where d f (x, ξ) = d ⊲ f (x, ξ) = [d, f (x, ξ)] ⊲ 1 and d ⊲ 1 = 0. Note that the action Mµν ⊲ f (x, ξ) in Eq. (8) is compatible with (6), (7) and Mµν ⊲ f (x, ξ)g(x, ξ) = Mµν f (x, ξ)g(x, ξ) ⊲ 1

(10)

The realization for Mµν in SH(x, ∂, ξ, q) is Mµν = xµ ∂ν − xν ∂µ + ξµ qν − ξν qµ

(11)

and now it is easy to verify Eqs.(4) - (7). Note that the Lorentz generators without the Grassmann-part (ξµ qν − ξν qµ ) cannot satisfy the condition (9). Usually in differential geometry vector field v = vµ ∂µ acts on a one-form ξβ = dxβ as a Lie derivative Lv ξβ = dLv xβ = dvβ . In our approach the action through Lie derivative is equivalent to the action of (vµ ∂µ + dvµ qµ ) ⊲ dxβ = dvβ and [vµ ∂µ + dvµ qµ , d] = 0. Our approach is more suitable for studying NC case (see section VI.). Hidden supersymmetry proposed in [53] could be interpreted as having origin in additional vectorlike Grassmann coordinates. The action of superspace realization of Lorentz generators (11) on physical superfields and possible physical consequences are still under consideration and will be presented elsewhere. 4

III.

κ-MINKOWSKI SPACE IN BICROSSPRODUCT BASIS

In κ-Minkowski space5 with deformed coordinates { xˆµ } we have [ xˆi , xˆ j ] = 0,

[ xˆ0 , xˆi ] = ia0 xˆi ,

(12)

where a0 is the deformation parameter. The deformed coproducts ∆ for momentum generators pµ and ˆ µν in bicrossproduct basis [10] are Lorentz generators M ∆p0 = p0 ⊗ 1 + 1 ⊗ p0 ,

∆pi = pµ ⊗ 1 + ea0 p0 ⊗ pi ,

ˆ i0 = M ˆ i0 ⊗ 1 + ea0 p0 ⊗ M ˆ i0 − a0 p j ⊗ M ˆ i j, ∆M

(13)

ˆ ij = M ˆ ij ⊗ 1 + 1 ⊗ M ˆ i j. ∆M

ˆ µν and pµ is called κ-Poincaré algebra where M ˆ µν generate undeformed Lorentz The algebra generated by M ˆ µν , pλ ] are given in [10]. Equations in algebra, pµ satisfy [pµ , pν ] = 0 and the commutation relations [ M (13) describe the coalgebra structure of the κ-Poincaré algebra and together with antipode and counit make ˆ xˆ, p) 7→ A( ˆ xˆ), where the κ-Poincaré-Hopf algebra. We have the action (for more details see [24]) ◮ : H( ˆ xˆ, p) is the algebra generated by xˆµ and pµ and A( ˆ xˆ, p) generated by xˆµ : ˆ xˆ) is a subalgebra of H( H( xˆµ ◮ gˆ ( xˆ) = xˆµ gˆ ( xˆ), pµ ◮ xˆν = −iηµν ,

pµ ◮ 1 = 0,

ˆ µν ◮ 1 = 0 M

(14)

ˆ µν ◮ xˆ λ = ηνλ xˆµ − ηµλ xˆν . M

Namely, using coproducts (13) and action (14) one can extract the following commutation relations between ˆ µν , pµ , and xˆ µ : M [p0 , xˆ µ ] = −iη0µ ,

[pk , xˆµ ] = −iηkµ + iaµ pk ,

ˆ µν , xˆλ ] = ηνλ xˆ µ − ηµλ xˆν − iaµ M ˆ νλ + iaν M ˆ µλ . [M

(15)

(16)

ˆ µν , pµ , and xˆµ in terms of undeformed xµ The realization corresponding to bicrossproduct basis for M and ∂µ is6 : xˆ(o) i = xi ,

ˆ (o) M i0

pµ = −i∂µ xˆ(o) 0 = x0 + ia0 xk ∂k , !
1 − Z ia0 2 2 2 1
ˆ (o) = xi ∂ j − x j ∂i , = xi ∂k − A Z − x0 + ia0 xk ∂k ∂i , M + sh ij ia0 2 ia0 2

(17)

where A = −ia0 ∂0 and Z = eA (for more details see [16] and [19]). 5 6

Greek indices (µ, ν, ...) are from 0 to 3, and Latin indices (i, j, ...) from 1 to 3. Summation over repeated indices is assumed. Here the superscript (o) denotes that the Lorentz generators and NC coordinates xˆ are realized only in terms of undeformed xµ and ∂µ .

5

IV.

BICOVARIANT DIFFERENTIAL CALCULUS

In the paper by Sitarz [11] there is a construction of a bicovariant differential calculus [51] on κMinkowski space compatible with Lorentz covariance condition (20), but with an extra one-form φ, which transforms as a singlet under the Lorentz generators. The algebra generated by xˆµ and one-forms ξˆµ , φ is closed in one-forms. ˆ xˆ µ ] = ξˆµ and satisfies ordinary Leibniz rule. The deformed exterior derivative is defined by dˆ 2 = 0, [d, ˆ µν and momentum In the bicovariant calculus it is also assumed that the coproduct for Lorentz generator M generator pµ is in the bicrossproduct basis (13), the action ◮7 is defined in (14) and it is extended to oneforms by ξˆµ ◮ 1 = ξˆµ ,

φ◮1=φ

ˆ µν ◮ φ = 0, pµ ◮ ξˆν = pµ ◮ φ = M

(18)

ˆ µν ◮ ξˆλ = ηνλ ξˆµ − ηµλ ξˆν . M

ˆ µν , ξˆλ ] and [pµ , ξˆν ]. In addition to From coproducts (13) and eq.(18) we can find commutation relations [ M Eqs. (15) and (16) we have ˆ µν , φ] = 0, [pµ ,ξˆν ] = [pµ , φ] = [ M

(19)

ˆ µν , ξˆλ ] = ηνλ ξˆµ − ηµλ ξˆν . [M The Lorentz covariance condition ˆ µν ◮ fˆ( xˆ, ξ)ˆ ˆ g( xˆ, ξ) ˆ = m ∆M ˆ µν ◮ fˆ ⊗ gˆ
M

(20)

ˆ = 0, ˆ µν , d] [M

(21)

ˆ M ˆ µν ◮ dˆ fˆ = d( ˆ µν ◮ fˆ) M

implies

ˆ fˆ] ◮ 1 and dˆ ◮ 1 = 0. where dˆ fˆ = dˆ ◮ fˆ = [d, Sitarz claims that the algebra8 between one-forms ξˆµ , φ and NC coordinate xˆµ that is compatible with (20) - (21) is given by [ xˆ0 , ξˆ0 ] = −a20 φ,

[ xˆµ , φ] = ξˆµ ,
[ xˆi , ξˆ j ] = −ia0 δi j ξˆ0 + ia0 φ , 7 8

Sitarz denotes this action with ⊲. The correspondence between algebra in [11] and (22) is

1 κ

[ xˆ0 , ξˆi ] = 0,

(22) [ xˆi , ξˆ0 ] = −ia0 ξˆi .

ˆ 0i , and Mi = ǫi jk M ˆ jk . = −ia0 , xµ = xˆµ , dxµ = ξˆµ , φ = φ, Ni = M

6

The realization for xˆµ is given in (17) and the realization for one-forms ξˆµ and φ that satisfies (22) can be given in terms of undeformed xµ , ∂µ and ξµ 9 . The realizations10 for one-forms and exterior derivative are  a2  ξˆ0 = ξ0 1 + 0  + ia0 ξk ∂k , ξˆk = ξk − ia0 ξ0 ∂k Z −1 , 2  Z −1 − 1 ia0  4 1
+  − ξk ∂k ,  = ∂2i Z −1 − 2 sh2 A , φ = −dˆ s = ξ0 ia0 2 2 a0

(23)

where we have denoted the exterior derivative for Sitarz’s case with dˆ s . ˆ is not isomorphic to Relations (22), (23) and (2) imply φ = −dˆ s and φ ⊲ 1 = 0, so that the algebra11 SA ˆ µν cannot be realized in terms SA⋆ . Also the problem with this construction is that the Lorentz generators M of xµ , ∂µ , ξµ and qµ in order to satisfy Lorentz covariance condition (20) which implies (21). Namely, if ˆ µν so that [ M ˆ µν , dˆ s ] = 0 is fulfilled, we take the realization (23) for dˆ s and just want to find realization for M ˆ µν do not satisfy the Lorentz algebra (4). then these M Furthermore, in [26] differential algebras D1 and D2 of classical dimension were constructed (avoiding ˆ µν does not commute with exterior the extra form φ), where all conditions were satisfied, except (16) and M derivative. All these arguments lead to a conclusion that for the fixed realization (17) for xˆµ , there is no ˆ µν that satisfies κ-Poincaré-Hopf algebra (13) and Lorentz covariance condition (20), (21). realization for M

V.

κ-MINKOWSKI SPACETIME FROM TWIST AND NC ONE-FORMS

In this section we will construct the noncommutative coordinates xˆµ , coproducts, and NC one-forms using the twist operator.

A. κ-Minkowski spacetime from twist

We start with an Abelian twist (see [47], [20], [36] and [22])
F = exp −A ⊗ xk ∂k ,

(24)

where A = ia∂ = −ia0 ∂0 . The bidifferential operator (24) satisfies all the properties of a twist (2-cocycle condition and normalization) and leads to noncommutative coordinates   xˆµ = m0 F −1 ⊲ (xµ ⊗ id) . 9

10 11

(25)

The algebra of undeformed operators is defined in Section II. and for ◮ action we have xµ ◮ 1 = xˆµ , ξµ ◮ 1 = ξˆµ , ∂µ ◮ 1 = 0 and qµ ◮ 1 = 0. For more details see [26] ˆ is generated by xˆµ , ξˆµ and φ, and the algebra SA⋆ is generated by xµ and ξµ but with ⋆-multiplication. The The algebra SA ˆ g( xˆ, ξ) ˆ ⊲ 1. star-product is defined by f (x, ξ) ⋆ g(x, ξ) = fˆ( xˆ, ξ)ˆ

7

It follows that for this twist we get a realization for xˆµ exactly as in (17). The twist given by Eq. (24) also leads to an associative star product   f (x) ⋆ g(x) = m0 F −1 ⊲ ( f ⊗ g) .

(26)

If we define the operators Mµν as Mµν = xµ ∂ν − xν ∂µ , then Mµν generate the undeformed Lorentz algebra, but their coproducts, obtained from the twist (24) do not close in the Poincaré-Hopf algebra. For this reason we consider the algebra igl(4), generated by ∂µ and Lµν = xµ ∂ν , which also has a Hopf algebra structure [25]. The coproducts of ∂µ and Lµν calculated as ∆∂µ = F ∆0 ∂µ F −1 , and analogously for Lµν , are ∆∂0 = ∆0 ∂0 , ∆Li j = ∆0 Li j ,

∆∂i = ∂i ⊗ 1 + eA ⊗ ∂i

(27)

∆L00 = ∆0 L00 + A ⊗ Lkk

∆Li0 = Li0 ⊗ 1 + e−A ⊗ Li0 ,

∆L0i = L0i ⊗ 1 + eA ⊗ L0i − ia0 ∂i ⊗ Lkk .

(28) (29)

Coproducts of the momenta pµ , obtained from (27) by expressing pµ in terms of ∂µ (pµ = −i∂µ ), coincide with the coproducts of momenta in the bicrossproduct basis (13) (see also [10]). On the other hand, coproducts of the Lorentz generators Mµν , calculated from Eqs. (28) and (29) as ∆Mµν = ∆Lµν − ∆Lνµ , are different from the ones in the bicrossproduct basis (13) (more precisely, ∆Mi0 is different), [25]. In this way we have constructed the igl(4)-Hopf algebra structure using twist F . We point out that in [27, 52] it is shown that for Lorentz generators in bicrossproduct basis (17) the twist ˆ µν and pµ . F gives the correct Hopf algebra structure (13) of κ-Poincaré algebra generated by M

B.

Noncommutative one-forms from twist

Our aim is to construct an exterior derivative dˆ and noncommutative one-forms ξˆµ with the following properties: dˆ 2 = 0,

ˆ xˆµ ] = ξˆµ [d,

{ξˆµ , ξˆν } = 0,

(30)

λ ˆ λ [ξˆµ , xˆν ] = Kµν ξλ , Kµν ∈C

[ξˆµ , xˆν ] − [ξˆν , xˆµ ] = iaµ ξˆν − iaν ξˆµ

(consistency condition),

(31) (32)

where we have introduced aµ = (a0 , ~0) so that (12) can be written in a unified way as [ xˆµ , xˆν ] = i(aµ xˆν − aν xˆµ ).

(33)

We want to find a realization of NC one-forms in terms of the undeformed algebra SH(x, ∂, ξ, q). If we   now calculate ξˆµ , by analogy to Eq. (25), as ξˆµ = m0 F −1 ⊲ (ξµ ⊗ id) , we get ξˆµ = ξµ , so that the LHS of 8

(32) equals 0, while the RHS gives iaµ ξˆν − iaν ξˆµ and the consistency condition is not fulfilled. Obviously we need to extend the twist defined in (24). We have shown that the bicovariant differential calculus á la Sitarz [11] could not be realized in terms of Heisenberg or super-Heisenberg algebra. In the next section we will propose a new version of bicovariant calculus compatible with igl(4)-Hopf algebra.

VI.

EXTENDED TWIST

Our main goal is to construct a twist so that our bicovariant calculus satisfies the following properties: 1. The bicovariant calculus has classical dimension, i.e. there is no extra form like φ. 2. The algebra between ξˆµ and xˆµ is closed in one-forms. 3. Generators Mµν satisfy the Lorentz algebra. ˆ = 0, which is sufficient condition for (20), (21). 4. The condition [Mµν , d] In order to satisfy all the requirements for ξˆµ and dˆ we define the extended twist   Fext = exp −A ⊗ (xk ∂k + ξk qk ) .

(34)

This twist leads to     −1 −1 ⊲ (x0 ⊗ id) = x0 + ia0 (xk ∂k + ξk qk ) ⊲ (xi ⊗ id) = xi , xˆ0 = m0 Fext xˆi = m0 Fext   −1 ⊲ (ξµ ⊗ id) = ξµ . ξˆµ = m0 Fext

(35) (36)

Although the realization of xˆ0 is changed with the addition of a term containing Grassmann variables, xˆµ still satisfy the same commutation relations Eq. (12), but the commutation relations between xˆµ and ξˆµ are no longer all equal to 0 [ξˆµ , xˆi ] = 0,

[ξˆ0 , xˆ0 ] = 0,

[ξˆi , xˆ0 ] = −ia0 ξˆi .

(37)

Inserting (37) into (32) shows that in this case the consistency condition and the requirement from (31) are ˆ satisfied. Note that xˆµ , ξˆµ , ∂µ and qµ generate the deformed super-Heisenberg algebra SH,which also has super-Hopf-algebroid structure. We now want to introduce an exterior derivative dˆ such that (30) is also fulfilled and gives rise to the same expression for ξˆµ as (36). It is easily shown that this is achieved with dˆ = ξ α ∂α = d. 9

(38)

(0) ˆ µν = M ˆ µν + Grassmann part For the exterior derivative d in (38), we wanted to constructed an operator M

by extending the realization in SH with the property that it commutes with exterior derivative, i.e. ˆ µν , d] = 0, but in doing so, we find that this operator does not satisfy the Lorentz algebra (4). Hence, [M exterior derivative (38) is not be compatible with κ-Poincaré-Hopf algebra in the bicrossproduct basis even if we consider the realizations in SH. Since ξˆµ are undeformed, their ⊲ and ◮ actions are the same as for ξµ . Our construction can be extended to forms of higher order in a natural way. E.g., the space of two-forms can be defined as the space generated by ξˆµ ∧ ξˆν . These two-forms then automatically satisfy ξˆµ ∧ ξˆν = ξµ ∧ ξν = −ξν ∧ ξµ = −ξˆν ∧ ξˆµ .

(39)

Now we define the extended ⋆-product with   −1 ⊲ f ⊗g . f (x, ξ) ⋆ g(x, ξ) = m0 Fext

(40)

For f (x) and g(x), functions of x only, (40) coincides with (26), and if f (ξ) and g(ξ) are functions of ξ only, their extended ⋆-product is just the ordinary multiplication, f (ξ) ⋆ g(ξ) = f (ξ)g(ξ). As before (see (26)), ˆ g( xˆ, ξˆ) ⊲ 1. the extended ⋆-product can be equivalently defined with the ⊲ action: f (x, ξ) ⋆ g(x, ξ) = fˆ( xˆ, ξ)ˆ In order to get a Lorentz i.e. igl(4, R) covariant action, generators of gl(4) also need to be extended. Lext µν ext are defined by Lext µν = xµ ∂ν + ξµ qν . It can be easily checked that Lµν , defined in this way, still satisfy the

igl(4) algebra, and furthermore, that they commute with d, [Lext µν , d] = 0, so that ext ˆ ext ext ˆ ˆ Lext µν ◮ ξλ = [Lµν , ξλ ] ◮ 1 = d[Lµν , xˆλ ] ◮ 1 = d(Lµν ◮ xˆλ ) = ξµ ηνλ .

(41)

The results for the coproducts, obtained from the extended twist, are ∆q0 = ∆0 q0 ,

∆qi = qi ⊗ 1 + (−)deg eA ⊗ qi

ext ∆Lext i j = ∆0 Li j ,

ext ext ∆Lext 00 = ∆0 L00 + A ⊗ Lkk

−A ext ⊗ Lext ∆Lext i0 , i0 = Li0 ⊗ 1 + e

ext A ext ext ∆Lext 0i = L0i ⊗ 1 + e ⊗ L0i − ia0 ∂i ⊗ Lkk .

(42) (43) (44)

The coproducts of ∂0 and ∂i , calculated in the same way, are again given by Eq. (27). ˆ The action of Lext µν on the product of xˆρ and ξσ , calculated in three different ways: ext ˆ ˆ ˆ ˆ ρ ◮ ([Lext (i) Lext µν , ξσ ] ◮ 1) µν ◮ xˆρ ξσ = [Lµν , xˆρ ] ◮ ξσ + x
ext ˆ
(ii) Lext ˆρ ξˆσ = Lext µν ◮ x µν(1) ◮ xˆρ Lµν(2) ◮ ξσ

ext ext ˆ (iii) Lext µν ◮ xˆρ ξσ = Lµν(1) ◮ xˆρ d Lµν(2) ◮ xˆσ ,

gives the same result, i.e., the action is in accordance with bicovariant calculus. One can easily show that neither the last equality, (iii), nor (41) would be satisfied had we used the ordinary definition of Lµν 10

(Lµν = xµ ∂ν ). Similar expressions can be written in terms of the ⊲ action and the extended ⋆ product. Hence, we have constructed igl(4)-Hopf algebra structure using the extended twist Fext that satisfies all the requirements for bicovariant calculus (listed in the beginning of this section 1-4). Note that the coproduts ext of Mµν = Lext µν − Lνµ calculated in this way differ from the one in bicrossproduct basis.

The action (41) could also be obtained using the ordinary definition of Lµν and promoting these generators to Lie derivatives. Using Cartan’s identity, one would get L xµ ∂ν ⊲ ξλ = d(L xµ ∂ν ⊲ xλ ) = d(xµ ηνλ ) = ξν ηνλ .

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However, there is a problem in this approach. Namely, promoting Lµν to Lie derivatives would again give the realization (17) for xˆµ and in the case of deformed one-forms it would give ξˆµ = ξµ , i.e., the consistency condition would not be fulfilled.

VII.

OUTLOOK AND DISCUSSION

We have shown that if the NC coordinates (12) are given only in terms of Heisenberg algebra H(x, ∂), then there is no realization of Lorentz generators compatible with all the requirements of bicovariant calculus [11]. Hence, if one wants to unify κ-Minkowski spacetime, κ-Poincaré algebra and differential forms it ˆ xˆ, ξˆ, ∂, q). This is explicitly done is crucial to embed them into κ-deformed super-Heisenberg algebra SH( for κ-deformed igl(4)-Hopf algebra using the extended twist. We choose a bicrossproduct basis just as one example, but similar constructions for other bases are also possible. In section VI we have constructed a bicovariant differential calculus compatible with κ-deformed igl(4)Hopf algebra. The question is whether it is possible to construct bicovariant calculus compatible with κ-Poincaré-Hopf algebra. One has to develop the notion of superphase space and its super-Hopf-algebroid structure. The idea is to generalize the method developed in [52] where we have analyzed the quantum phase space, its deformation and Hopf-algebroid structure. In [52] we have also constructed κ-PoincaréHopf algebra from twist. We present the realization of κ-Poincaré algebra compatible with bicovariant differential calculus in the bicrossproduct basis. Starting with the extended realization of xˆµ , xˆi = xi ,

xˆ0 = x0 + ia0 (xk ∂k + ξk qk )

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and demanding the Lorentz algebra (4), (13), and (16) we find the realization for the Lorentz generators: ˆ i j = xi ∂ j − x j ∂i + ξi q j − ξ j qi M ˆ i0 = M ˆ (0) + ξi (q0 Z 2 + ia0 qk ∂k ) − [ξ0 qi + ia0 (ξk qk ∂i + ξk ∂k qi )] M i0 11

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ˆ µν commutes with exterior ˆ (0) is given in (17). The requirement that the Lorentz generators M where M i0 ˜ i.e. [d, ˜ M ˆ µν ] = 0 is fulfilled for derivatives d, " ! # 1 shA ia0 2 −1 ˜d = ξ0 ∂ Z + + ξ j∂ jZ a2 ia0 2 i 1 + 20 

(48)

˜ xˆµ ]. We are working on generalizing the results in [52] The corresponding one-forms are defined as ξ˜µ = [d, in order to construct the extended twist operator within the superphase space which will provide the correct κ-Poincaré-Hopf algebra compatible with bicovariant calculus. All the details of this construction will be presented elsewhere. Our main motivation for studying these problems is related to the fact that the general theory of relativity together with uncertainty principle leads to NC spacetime. In this setting, the notion of smooth spacetime geometry and its symmetries are generalized using the Hopf algebraic approach. Further development of the approach presented in this paper will lead to possible application to the construction of NC quantum field theories (especially electrodynamics and gauge theories), quantum gravity models, particle statistics, and modified dispersion relation.

Acknowledgment We would like to thank Peter Schupp for useful comments. This work was supported by the Ministry of Science and Technology of the Republic of Croatia under contract No. 098-0000000-2865. R.Š. gratefully acknowledges support from the DFG within the Research Training Group 1620 “Models of Gravity”.

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